Across(to or from the other side)- we saw them walking across the street.
Zebra crossing is across the road
we are walking across the river ( means we are walking in the river physically inside it and connecting the two edge of that particular river)
(which indicates the connectivity)
over - the bridge is over the river
we are walking over the river ( means we are walking on the bridge over the river and the river is under the bridge)
(it is not mandatory to establish physical touch)